Dragoş Ştefan

A categorical approach to cyclic duality

The aim of this paper is to provide a unifying categorical framework for the many examples of para-(co)cyclic modules arising from Hopf cyclic theory. Functoriality of the coefficients is immediate in this approach. A functor corresponding to Conness cyclic duality is constructed. Our methods allow, in particular, to extend Hopf cyclic theory to (Hopf) bialgebroids.

The cohomology ring of the 12-dimensional Fomin–Kirillov algebra

Abstract The 12-dimensional Fomin–Kirillov algebra FK 3 is defined as the quadratic algebra with generators a , b and c which satisfy the relations a 2 = b 2 = c 2 = 0 and a b + b c + c a = 0 = b a + c b + a c . By a result of A. Milinski and H.-J. Schneider, this algebra is isomorphic to the Nichols algebra associated to the Yetter–Drinfeld module V , over the symmetric group S 3 , corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring Ext FK 3 ⁎ ( k , k ) , showing that it is a polynomial ring S [ X ] with coefficients in the symmetric braided algebra of V . As an application we also compute the cohomology rings of the bosonization FK 3 # k S 3 and of its dual, which are 72-dimensional ordinary Hopf algebras.

DEFORMATION COHOMOLOGY FOR YETTER-DRINFEL'D MODULES AND HOPF (BI)MODUL ES

If A is a bialgebra over a field k, a left-right Yetter-Drinfel’d module over A is a k-linear space M which is a left A-module, a right A-comodule and such that a certain compatibility condition between these two structures holds. YetterDrinfel’d modules were introduced by D. Yetter in [18] under the name of “crossed bimodules” (they are called “quantum Yang-Baxter modules” in [5]; the present name is taken from [10]). If A is a finite dimensional Hopf algebra then the category of left-right Yetter-Drinfel’d modules is equivalent to the category of left modules over D(A), the Drinfel’d double of A (see [6], [9]), even as braided tensor categories, and also to the category of Hopf bimodules over A (see [1], [11], [12], [17]). An important class of examples occurs as follows: if M is a finite dimensional vector space and R ∈ End(M ⊗ M) is a solution to the quantum Yang-Baxter equation, then the so-called “FRT construction” (see for instance [2]) associates to R a certain bialgebra A(R), and M becomes a left-right Yetter-Drinfel’d module over A(R) (see [5], [9]). In this paper we introduce a cohomology theory for left-right Yetter-Drinfel’d modules. If A is a bialgebra and M,N are left-right Yetter-Drinfel’d modules over A, we construct a double complex {Y (M,N)} whose total cohomology is the desired cohomology H(M,N). For M = N = k, this cohomology

Descent theory and Amitsur cohomology of triples

Abstract For a given triple (monad) U : C → C in the category C , we develop a theory of descent for U. We start by introducing the basic constructions associated to a triple: descent data, symmetry operators, and flat connections. The main result of this section asserts that the sets of these objects are bijectively equivalent. Next we construct a monoidal category C (U) such that U is an algebra in C (U) . If C is abelian, we define Amitsur cohomology of U with coefficients in a functor F : C (U)→ D . As an application of this construction, in the case where U is faithfully exact, we describe those morphisms that descend with respect to U. In the last part of the paper we classify all U-forms of a given object C 0 ∈ C . We show that there is a one-to-one correspondence between the set of equivalence classes of U-forms and a certain noncommutative Amitsur cohomology. Let A/B be an extension of associative unitary rings and let C be the category of right B-modules. Then (−)⊗ B A : C → C is a triple which is faithfully exact if and only if the extension A/B is faithfully flat. Specializing our results to this particular setting, we recover faithfully flat descent theory for extensions of (not necessarily commutative) rings.

Examples of Para-cocyclic Objects Induced by BD-Laws

In a recent paper (Böhm and Stefan, Commun Math Phys 282:239–286, 2008), we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple—consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate (‘locally braided’) morphisms of monads.

Decomposition of hochschild homology of hopf galois extensions

Let H be a Hopf algebra over a field k, and let A/B be a Hopf Galois extension. We shall construct a = H/[H, H] comodule structure on the Hochschild homology of A with coefficients in a Hopf bimodule M. Then, for a subcoalgcbra C , we construct certain functors HC(A, –) from Hopf bimodules to k-spaces and we set up a spectral sequence for describing them

On Koszulity of finite graded posets

In this paper, we continue our research on Koszul rings, started in [P. Jara, J. Lopez-Pena and D. Ştefan, Koszul pairs and applications, to appear in J. Noncommut. Geom., http://arxiv.org/pdf/1011.4243.pdf]. In Theorem 1.9, we prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of TornA(R,R). As an application of these characterizations, we investigate the Koszulity of the incidence rings for finite graded posets, see Theorems 2.8 and 2.9. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets, whose incidence rings are Koszul). Specific examples of Koszul posets are included.

Factorizable enriched categories and applications

We define the twisted tensor product of two enriched categories, which generalizes various sorts of ‘products’ of algebraic structures, including the bicrossed product of groups, the twisted tensor product of (co)algebras and the double cross product of bialgebras. The key ingredient in the definition is the notion of simple twisting systems between two enriched categories. To give examples of simple twisted tensor products we introduce matched pairs of enriched categories. Several other examples related to ordinary categories, posets and groupoids are also discussed.

Further Properties and Applications of Koszul Pairs

Koszul pairs were introduced in (arXiv:1011.4243) as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in (arXiv:1605.05458). As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite R-rings, we show that a graded R-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.

Cleft Comodule Categories

We define crossed product categories and we show that they are equivalent with cleft comodule categories. We also prove that a comodule category is cleft if and only if it is Hopf–Galois and has a normal basis. As an application we show that the category of Hopf modules over a cleft linear category and the category of modules over the coinvariant subcategory are equivalent.